To simplify the expression \( 80x^3y^2 \), we can factor it down.
- Factor out the coefficients and variables:
- \( 80 = 16 \times 5 \)
- \( x^3 = x^2 \cdot x \)
- \( y^2 = y^2 \) (it stays the same)
Thus, we can write:
\[ 80x^3y^2 = 16 \cdot 5 \cdot x^2 \cdot x \cdot y^2 \]
- Grouping components: If we want to express in terms of the choices provided, we can also breakdown \( 80x^3y^2 \):
Options given:
- \( 4x^2y^2\sqrt{5x} \)
- \( 4xy\sqrt{5x} \)
- \( 5x^2y^2\sqrt{4x} \)
- \( 5xy\sqrt{4x} \)
Approaching the square elements:
We don't have a direct simplification that matches any of the provided forms without additional modifications. However, let's see how close we can get to one of these forms via square roots.
\[ 80x^3y^2 = 16 \cdot 5 \cdot x^2 \cdot x \cdot y^2 = (4x^2y^2) \times (20x) \]
Then express \( 20x \) specifically as \( 4\sqrt{5x} \times 5 \).
- Calculation with square roots: Let's see if we can manipulate and simplify with roots.
Using \( 20 = 4\sqrt{5x} \):
\[ 80x^3y^2 = 4x^2y^2 \cdot (5\cdot 4\sqrt{5x}) = 4x^2y^2\sqrt{20x} \]
This does not fit any provided choices exactly either.
Conclusion
From these steps, it's clear that the simplifications provided do not precisely match the expression \( 80x^3y^2 \) simplifications in various presented forms.
However, the expression \( 5x^2y^2\sqrt{4x} \) could imply a multiple.
So in conclusion, if you have options as choices to select, examine the definitions closely to check values; however, the best answer from given choices appears to be: \( 4x^2y^2\sqrt{5x} \) based on proximity and transformations but would still yield differing input from what was observed. It really depends on how you are looking or plotting directly against input!!